Question

Explain Monte Carlo Sampling? Under what circumstances, can it be used? Elaborate on the application and limitations related to this sampling?

Answer 1

Monte Carlo simulation is named after the city of Monte Carlo in Monaco, which is famous for gambling such as roulette, dice, and slot machines. Since the simulation process involves generating chance variables and exhibits random behaviours, it has been called the Monte Carlo simulation

Monte Carlo simulation has been applied to diverse problems ranging from the simulation of complex physical phenomena such as atom collisions to the simulation of traffic flow and Dow Jones forecasting. Monte Carlo is also suitable for solving complex engineering problems because it can deal with a large number of random variables, various distribution types, and highly nonlinear engineering models

Sampling on input random variables

The purpose of sampling on the input random variables 12 (,,,) nX XX=X L is to generate samples that represent distributions of the input variable from their cdfs () (1,2,,) i Xi Fxin = L . The samples of the random variables will then be used as inputs to the simulation experiments

Step 1 – Generating random variables that are uniformly distributed between 0 and 1

Step 2 – Transforming [0, 1] uniform variables into random variables that follow the given distributions

the features of Monte Carlo simulation are summarized as follows.

1) Monte Carlo simulation is easy to use for engineers who have only limited working knowledge of probability and statistics.

2) Monte Carlo simulation is feasible to use for virtually any performance functions and distributions.

3) Monte Carlo simulation is computationally robust; with sufficient number of simulations, it can always converge.

4) The problem dimension (the number of random variables) does not affect the accuracy of Monte Carlo simulation as indicated in Eq. 9.14. This feature is beneficial to large scale engineering problems.

5) For reliability analysis, Monte Carlo simulation is generally computationally expensive. The higher the reliability is, the larger the simulation size is needed.

Because of the accuracy, Monte Carlo simulation is widely used in 1) engineering applications where the model evaluations (deterministic analyses) are not computationally expensive and 2) validating other methods. However, due to its computational inefficiency, Monte Carlo simulation is not commonly used for problems where deterministic analyses are expensive